# Video: US-SAT05S4-Q20-659123709815

Consider the function 𝑓(𝑥, 𝑦) = (𝐾𝑥³)/𝑦. If 𝐾 is a constant in the definition of the function 𝑓 and 𝑓(𝑚, 𝑛) = 3, what is the value of 𝑓(2𝑚, 3𝑛)?

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### Video Transcript

Consider the function 𝑓 of 𝑥, 𝑦 equals 𝐾 times 𝑥 cubed over 𝑦. If 𝐾 is a constant in the definition of the function 𝑓 and 𝑓 of 𝑚, 𝑛 equals three, what is the value of 𝑓 of two 𝑚, three 𝑛?

At first, it might not seem like we have enough information to solve this problem. But let’s write down what we know: 𝑓 of 𝑥, 𝑦 equals 𝐾 times 𝑥 cubed over 𝑦 and 𝑓 of 𝑚, 𝑛 equals three. We could also say that 𝑓 of 𝑚, 𝑛 equals 𝐾 times 𝑚 cubed over 𝑛. And since we’re interested in 𝑓 of two 𝑚, three 𝑛, we can say that 𝑓 of two 𝑚, three 𝑛 will equal 𝐾 times two 𝑚 cubed over three 𝑛. We can distribute this cube and say two cubed times 𝑚 cubed. So we can say that 𝑓 of two 𝑚, three 𝑛 equals 𝐾 times two cubed times 𝑚 cubed times three 𝑛.

The 𝐾 stays the same. Two cubed equals eight. 𝑚 cubed equals 𝑚 cubed. And then in our denominator, we have three times 𝑛. Can we regroup 𝐾 times eight times 𝑚 cubed over three times 𝑛 to use the information from 𝑓 of 𝑚, 𝑛? If we take out the eight-thirds, we could say that 𝑓 of two 𝑚, three 𝑛 is equal to eight-thirds times 𝐾𝑚 cubed over 𝑛, which is 𝑓 of 𝑚, 𝑛. And we know that that equals three. That means 𝑓 of two 𝑚, three 𝑛 is equal to eight-thirds times three. The three in the numerator and the three in the denominator cancel out, leaving us with eight.

Based on the given information, we can say that 𝑓 of two 𝑚, three 𝑛 equals eight.